252 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



the 3 n varying coordinates and the time, involving also the masses and the 6 n initial 

 data above mentioned ; the discovery of which relations would be (as we have said) 

 the general solution of the general problem of dynamics. We have, therefore, at 

 least reduced that general problem to the search and differentiation of a single func- 

 tion V, which we shall call on this account the characteristic function of motion of 

 a system ; and the equation (A.), expressing the fundamental law of its variation, we 

 shall call the equation of the characteristic function, or the law of varying action. 



3. To show more clearly that the action or accumulated living force of a system, 

 or in other words, the integral of the product of the living force by the element of the 

 time, may be regarded as a function of the &n-\- \ quantities already mentioned, 

 namely, of the initial and final coordinates, and of the quantity H, we may observe, 

 that whatever depends on the manner and time of motion of the system may be con- 

 sidered as such a function ; because the initial form of the law of living foi'ce, when 

 combined with the 3 n known or unknown relations between the time, the initial data, 

 and the varying coordinates, will always furnish ?>n-\- 1 relations, known or unknown, 

 to connect the time and the initial components of velocities with the initial and final 

 coordinates, and with H. Yet from not having formed the conception of the action 

 as a function of this kind, the consequences that have been here deduced from the 

 formula (A.) for the variation of that definite integral, appear to have escaped the 

 notice of Lagrange, and of the other illustrious analysts who have written on theo- 

 retical mechanics ; although they were in possession of a formula for the variation of 

 this integral not greatly differing from ours. For although Lagrange and others, in 

 treating of the motion of a system, have shown that the variation of this definite inte- 

 gral vanishes when the extreme coordinates and the constant H are given, they appear 

 to have deduced from this result only the well known law of least action ; namely, 

 that if the points or bodies of a system be imagined to move from a given set of initial 

 to a given set of final positions, not as they do nor even as they could move consist- 

 ently with the general dynamical laws or differential equations of motion, but so as 

 not to violate any supposed geometrical connexions, nor that one dynamical relation 

 between velocities and configurations which constitutes the law of living force ; and 

 if, besides, this geometrically imaginable, but dynamically impossible motion, be made 

 to differ infinitely little from the actual manner of motion of the system, between the 

 given extreme positions ; then the varied value of the definite integral called action, 

 or the accumulated living force of the system in the motion thus imagined, will differ 

 infinitely less from the actual value of that integral. But when this well known law 

 of least, or as it might be better called, of stationary action, is applied to the determi- 

 nation of the actual motion of a system, it serves only to form, by the rules of the 

 calculus of variations, the differential equations of motion of the second order, which 

 can always be otherwise found. It seems, therefore, to be with reason that Lagrange, 

 Laplace, and Poisson have spoken lightly of the utility of this principle in the 

 present state of dynamics. A different estimate, perhaps, will be formed of that 



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